Continuum mechanics 0. Introduction and motivation What is ...

Continuum mechanics
0. Introduction and motivation
Aleš Janka
office Math 0.107
ales.janka@unifr.ch
http://perso.unifr.ch/ales.janka/mechanics
September 22, 2010, Université de Fribourg
Continuum mechanics
What is Continuum mechanics?
0. Introduction
Continuum mechanics = domain of physics and engineering describing:
elasticity and plasticity of solids and
dynamics of fluids (liquids or gases).
Continuum mechanics
0. Introduction

What is a continuum?
A continuum = a physical object with mass which can be mapped
onto points of a subdomain Ω ⊂ IR 3 . Mass distribution in Ω is
supposed to be continuous:
for any subdomain ω ⊂ Ω, the mass m ω of ω is calculated by
∫
m ω = dm
and small changes in the size of ω produce small changes in m ω .
ω
What is a continuum?
Continuum mechanics
0. Introduction
Modeling objects as continua neglects atomic, molecular and
crystal structure of mass.
The continuum approach is nevertheless a good approximation on
length scales much greater than the molecular scale.
What does continuum mechanics do?
Applies fundamental physical laws (conservation of mass,
momentum and energy, force equilibrium . . . ) to continua to
derive differential equations describing their behavior.
Information about the particular material of the continua is added
through an empiric constitutive relation / law.
Continuum mechanics
0. Introduction

Continuum mechanics in practice
.. in engineering: crash-tests
Continuum mechanics
Continuum mechanics in practice
.. in engineering: crash-tests
0. Introduction
Continuum mechanics
0. Introduction

Continuum mechanics in practice
.. in engineering: crash-tests
Continuum mechanics
Continuum mechanics in practice
.. in engineering: aeronautics
0. Introduction
Continuum mechanics
0. Introduction

Continuum mechanics in practice
.. in engineering: aeronautics
Continuum mechanics
Continuum mechanics in practice
.. weather forecasts
0. Introduction
Meteosuisse forecast for Sep 22, 2010, temperatures
Continuum mechanics
0. Introduction

Continuum mechanics in practice
.. in natural sciences: formation of galaxies (fluid dynamics)
Continuum mechanics
0. Introduction
Continuum mechanics in practice
.. in natural sciences: formation of galaxies (fluid dynamics)
Continuum mechanics
0. Introduction

Continuum mechanics in practice
.. in medicine (bone and tissue mechanics, blood flow)
from Arbenz, van Lenthe, Mennel, Müller and Sala: A scalable multi-level
preconditioner for matrix-free µ-finite element analysis of human bone
structures, Int. J. Numer. Meth. in Engineering 73 (2008), pp. 927–947
Continuum mechanics
0. Introduction
Continuum mechanics in practice
.. in medicine (bone and tissue mechanics, blood flow)
from Arbenz, van Lenthe, Mennel, Müller and Sala: A scalable multi-level
preconditioner for matrix-free µ-finite element analysis of human bone
structures, Int. J. Numer. Meth. in Engineering 73 (2008), pp. 927–947
Continuum mechanics
0. Introduction

Continuum mechanics in practice
.. in biology (tissue mechanics and growth)
from Schmundt et al. 2006
Programme
Continuum mechanics
0. Introduction
Kinematic description of a continuum: deformation and
motion of Ω.
Mechanical equilibria and conservation laws.
Constitutive laws of materials: elastic and visco-elastic
materials, Newtonian fluids.
Typical problems of continuum mechanics: analytical and
numerical solution of elasto-statics/dynamics, compressible
and incompressible elasticity, Newtonian fluids.
Continuum mechanics
0. Introduction

Necessary mathematical techniques
Mechanical state and properties of a continuum are
independent of the choice of a coordinate system.
We will introduce and use tensor calculus: covariant and
contravariant tensors and basic tensor operations, tensor fields in
euclidean space, derivatives of tensors.
Solutions of differential equations will be calculated analytically
(on simple problems) or numerically: we will (re)-introduce the
basics of a finite element method.
Continuum mechanics
0. Introduction
The beauty of simple analytical formulas: rubber baloon
Great deal of understanding through a simple toy model
Inflate a rubber party-balloon with an internal gas pressure p
Initially, the baloon stretches to two different diameters, why?
Continuum mechanics
0. Introduction

The beauty of simple analytical formulas: rubber baloon
Great deal of understanding through a simple toy model
Force equilibrium on the cut:
σ
t
r
r0
r
p
|F σ | = |F p |
2πrtσ = πr 2 p
elastic stress: σ = Eε = 2πr − 2πr 0
2πr 0
E
rubber incompressibility: 4πr 2 t = 4πr 2 0 t 0
p(r) = 2E t 0 r 2 0
r 3 r − r 0
r 0
Continuum mechanics
0. Introduction
The beauty of simple analytical formulas: rubber baloon
Great deal of understanding through a simple toy model
Force equilibrium on the cut:
|F σ | = |F p |
2πrtσ = πr 2 p
elastic stress: σ = Eε = 2πr − 2πr 0
2πr 0
rubber incompressibility: 4πr 2 t = 4πr 2 0 t 0
p(r) = 2E t 0 r 2 0
r 3 r − r 0
r 0
E
Continuum mechanics
0. Introduction

The beauty of mathematical analysis: singularities
Why it breaks always at a kink?
von Mieses stress: indicator of plastic deformation and rupture
Mathematical analysis of the solution of elasticity equations
predicts, that rupture occurs in re-entrant corners (kinks)!
Continuum mechanics
0. Introduction
The beauty of mathematical analysis: (in)stability
Buckling phenomenon
The nightmare of civil engineers and architects
Continuum mechanics
0. Introduction

The beauty of mathematical analysis: (in)stability
Buckling phenomenon
The nightmare of civil engineers and architects
Continuum mechanics
0. Introduction
The beauty of mathematical analysis: (in)stability
Buckling phenomenon
The nightmare of civil engineers and architects
Continuum mechanics
0. Introduction

The beauty of mathematical analysis: (in)stability
Buckling phenomenon
But it can also be exploited to our advantage!
shock absorber for road safety
Continuum mechanics
0. Introduction